It is a fact though that tenue de compte (the exact French translation of "bookkeeping") exists but is hardly ever used outside the banking world and what a French person will read as "frais de tenue de comptes" will be listed as "bank account fees" on a bank statement written in English.

This could lead us to think that the difference between "accountancy" and "bookkepping" is a cultural one and if you look deeper into this we can see that the difference is not that obvious to the English speaking world and sometimes needs to be explained.

There is often a misconception that bookkeeping and accounting are the same thing. While they both work to assist you with your finances, there are some important distinctions between the tasks of a bookkeeper and an accountant. [source]

Pierre Labardin, a lecturer in management and Marc Nikitin, professor in management, wrote a paper about the history of the word comptabilité from a linguistic and historical perspective. We learn in this paper that the merging of the words "bookkeeping" and "accounting" is fairly recent in French history. In their abstract they write:

In the French language, the word comptabilité (accounting) has gradually replaced tenue des livres (bookkeeping) around the beginning of the 19th century. After that, adjectives were
progressively added (commercial accounting, industrial accounting, general accounting, auxiliary accounting, etc.), in keeping with the division of labour in newly created accounting departments.

The first part of the paper is called I. De la tenue des livres à la comptabilité (From bookkeeping to accountancy).

Accounting is the Latinate version of bookkeeping. As is the case with many other English words, this is, too, an example of a terminological doublet so characteristic of the English language (e.g. commence vs. begin, love vs. amour, etc.)

As is known, the Latinate terms within these lexical doublets tend to be used in more formal registers than its Anglo-Saxon equivalents.

Hence it is accounting, and not the native bookkeeping, that has given the name for the science of keeping financial records.

Thank you for your answer. I am not familiar with the discipline, but websites such as this one say that accounting is a broader discipline. Basically, bookkeeping is only to record transactions and is the first step to accounting. So they don't hold roughly the same meaning as would commence and begin, to take your example.
– MinaApr 1 '17 at 9:58

Checking money.stackexchange.com where they have separate tag for bookkeeping ("the process of recording business data in a prescribed format. It often forms the first step in the accounting procedures of a company") and accounting might show you that the two terms do not cover the same scope in English.
– LaureApr 1 '17 at 16:12